FM for Actuaries

112 CHAPTER 4
4.2 One-Period Rate of Return
For the purpose of performance comparison and reporting, the rate of return of a
fund is calculated on a regular basis, such as quarterly or annually. Theoretically,
the methodology of the IRR may be used with the modification that the “project”
terminates at the end of the reporting period. Thus, the last cash flow C n in (4.1) is
set equal to the negative of the end-of-period fund value (the project terminates by
withdrawing the end-of-period balance). In practice, this method is rarely used, due
to the complexity in solving for the IRR numerically, as well as the problem of multiple
roots. As there are typically multiple fund injections and withdrawals within
the reporting interval, the cash flows change signs and the problem of multiple root
is prevalent.
We now discuss methods of calculating the return of a fund over a 1-period
interval. The methodology adopted depends on the data available. We start with the
situation where the exact amounts of fund withdrawals and injections are known,
as well as the time of their occurrence. For illustration, we consider a 1-year period
with initial fund amount B 0 (equal to C 0 ). Cash flow of amount C j occurs at time t j
(in fraction of a year) for j =1, ··· ,n,where0 <t 1 < ··· <t n < 1. As before,
we adopt the convention that C j is positive for cash injection and negative for cash
withdrawal. Note that C j are usually fund redemptions and new investments, and
do not include investment incomes such as dividends and coupon payments.
Denoting the fund value before and after the transaction at time t j by Bj
B and
Bj A, respectively, we have BA j = Bj B + C j for j =1, ··· ,n. The difference
between Bj
B and Bj−1 A , i.e., the balance before the transaction at time t j and after
the transaction at time t j−1 , is due to investment incomes, as well as capital gains
and losses. To complete the notations, let the fund balance at time 1 be B 1 ,and
define Bn+1 B = B 1 and B0
A = B 0 (this notation will allow us to express the gross
return as (4.4) below). Figure 4.1 illustrates the time diagram of the fund balances
and the cash flows.
We now introduce two methods to calculate the 1-year rate of return. These are
the time-weighted rate of return (TWRR) and the dollar-weighted rate of return
(DWRR). 1 To compute the TWRR we first calculate the return over each subinterval
between the occurrences of transactions by comparing the fund balances just
before the new transaction to the fund balance just after the last transaction. If we
denote R j as the rate of return over the subinterval t j−1 to t j ,wehave
1+R j = BB j
Bj−1
A , for j =1, ··· ,n+1. (4.4)
1 The dollar-weighted rate of return is also called the money-weighted rate of return.